mSKR wrote:
IanStewart When I see 3 equations, and 3 variables, I usually check manually that information is not duplicate in multiple equations. E.g. Equations are unique with each other.
Please share your thoughts how do you make a quick decision on seeing
n equations with
n variables.
In general, there's no quick way to make a decision about 3 (or more) equations in 3 (or more) unknowns. As soon as you get beyond two equations/two unknowns, things are very complicated, and the standard techniques involve matrix algebra (a subject you normally learn in the first year of an undergraduate math degree). Those techniques aren't quick, and they're way beyond the scope of the GMAT.
So in general, if you see three equations in three unknowns on the GMAT, the only way to know if you'll get one solution, or more than one, is to solve. There's no instant way to tell if one equation duplicates the information from one or both of the others, because one equation can be a combination of the other two. For example if you have these equations:
a + b = 5
a + c = 3
2a + b + c = 8
the third equation is the sum of the first two. So it's not new information, and these equations will have infinitely many solutions. Maybe that is a bit obvious, but you could instead have
a + b = 5
a + c = 3
2a + 5b - 3c = 16
and now it's not obvious at all that the third equation is a combination of the first two, but it is (if you multiply the first equation by 5, and the second by 3, and subtract, you get the third equation). Here again you'd have infinitely many solutions, but the only practical way to discover that is by solving the equations and seeing what happens. And with GMAT equations, that's going to be faster than any of the techniques you'd learn in advanced math.
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